This section provides a physical interpretation of equations 3.7, which
will lead to a
deeper understanding of the mechanics of inversion.

The causal Green's function gc and the
acausal Green's ga function for a point source in a 3-D
homogeneous medium are given by:

=

=

(3.10)

where c is the medium velocity, ts denotes the source initiation
time, t denotes the listening time, and
.
A causal Green's function is typically used for forward modeling,
while the acausal (waves are propagating prior to the source time) Green's function
is used for backward propagation.

Note that these Green's functions are time invariant, which means
that their values depend on the time lag between the source time
ts and listening time t. This is useful, because it means that
a CSG collected in May should look the same as one collected in July
for the same recording geometry and experimental site. In general, the Green's function for
acoustic fields are time-invariant in an arbitrary velocity model (Morse and Feshbach, 1953).

Figure 3.5 shows that these Green's functions describe
either a backward or forward light cone, where the apex of the cone
kisses the
source point at the time ts. For a buried
point source each cone intersects the surface plane z=0 along an hyperbola.
The causal Green's function emulates exploding wavefronts from a point "source",
while the acausal Green's function emulates imploding
wavefronts from a point "sink".

Taking the Fourier transform of
equation 3.11 w/r to the
variable says:

=

=

or,
.
This relation between the acausal and causal Green's functions is
valid for arbitrary elastic media (Morse and Feshbach, 1953).

Figure 3.5:
Forward and backward light cones due to a buried point source.